Physics 207, Lecture 23, Nov. 22. Agenda: Catch up Chapter 18, Superposition and Standing Waves Superposition Interference Standing Waves Nodes, Anti-nodes . Assignments: Problem Set 9 due Tuesday, Dec. 5, 11:59 PM Ch. 18: 3, 18, 30 , 40, 58
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Ch. 18: 3, 18, 30, 40, 58
What does C(t) =A(t)+ B(t) look like ?
A(x,t)=A cos(kx–wt)B(x,t)=A cos(kx–wt+f)
Amplitude = 2A cos (f/2)
Phase shift = f / 2
(the other has a different phase)
will add constructively
will add destructively
CONSTRUCTIVEINTERFERENCESuperposition & Interference
2 = 1.15 x 1.
linear in x !
x(t) = B sin(t)+C cos(t)
yA(x,t)=A cos(k1x–w1t)yB(x,t)=A cos(k2x–w2t)
And let x=0, y=yA+yB = 2A cos[2p (f1 – f2)t/2] cos[2p (f1 + f2)t/2]
and |f1 – f2| ≡ fbeat = = 1 / Tbeat
The frequency difference was the samefor both pairs of waves.
Need more information.
Path (or x) phase = 2p (path/ l) (modulo 2p )
with n = 0, 1, 2, …
In phase sources separated by a distance d
Sound waves interfere, just like transverse waves do. The resulting wave (displacement, pressure) is the sum of the two (or more) waves you started with.
Optical Y Splitter
A Crystal with Line Defect Acting
as a Waveguide:
Si (n=3.4); Period A = 0.58mm;
Filling Factor = 5/16;
Excitation l = 1.55mm
Light turning a corner
The distances AD and BCD have equal transit times so the sound waves will be in phase. The only need is for AB = 1 wavelength
AB = l AD = BC+CD = BC + (h2 + (d/2)2)½ = d
AC = AB+BC = l +BC = (h2 + d/22)½
Eliminating BC gives l+d = 2 (h2 + d2/4)½
l + 2ld + d2 = 4 h2 + d2
1 + 2d = 4 h2 / l d = 2 h2 / l – ½
= 7.5 m
Because the ground is more dense than air there will be a phase change of p and so we really should set AB to l/2 or 0.5 m.
C’(x,t) = 2A sin(2px/l) sin(wt)
These are “standing waves”.
a bound string (length L)
C(0,t) = C(L,t) = 0 if
L = n l/2 l = 2 L/n
C’(x,t) = 2A sin(p n x/L) sin(wt)
A combination wave composed of the 1st harmonic and the third harmonic.
What makes instruments unique is the combination of harmonics produced by the different instruments.
Flutes produce primarily the 1st harmonic
They have a very pure tone
Oboes produce a broad range of harmonics and sound very differentMusic
Three ways to make sound harmonics produced by the different instruments.
Vibrate a string
Vibrate an air column
Vibrate a membraneMusical Instruments
Violin, viola, cello, string bass harmonics produced by the different instruments.
All vibrate a structure to “amplify” the soundVibrating Strings
Vibrating Air Columns
Open at one end:
Pressure AntiNode at closed end
Displacement Node at closed end
l = 4 L / n n = 1,3,5…
Open at both ends:
Pressure(speed) Node at ends
Displacement AntiNode at ends
l = 2 L / nn = 1,2,3..
A 0.9 m organ pipe (open at both ends) is measured to have it’s first harmonic (i.e., its fundamental) at a frequency of 382 Hz. What is the speed of sound (refers to energy transfer) in the pipe?
f = 382 Hzandf l = vwith l = 2 L / n(n = 1)
v = 382 x 2(0.9) m v = 687 m/s
Recall: f l = v
(A) Increases (B) Same (C) Decreases
Ch. 18: 9, 17, 21, 39, 53a (tentative)